How many ways can 3 different prizes be awarded to 5 students?

How many ways can 3 different prizes be awarded to 5 students?

Correct Answer: 60

Step 1: Understand that we have 3 different prizes to give out.
Step 2: Recognize that there are 5 students who can receive these prizes.
Step 3: Realize that the order in which we award the prizes matters because they are different.
Step 4: Use the formula for permutations, which is P(n, r) = n! / (n - r)!, where n is the total number of items (students) and r is the number of items to choose (prizes).
Step 5: In this case, n = 5 (students) and r = 3 (prizes).
Step 6: Calculate P(5, 3) = 5! / (5 - 3)! = 5! / 2!.
Step 7: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step 8: Calculate 2! = 2 × 1 = 2.
Step 9: Now divide: 120 / 2 = 60.
Step 10: Conclude that there are 60 different ways to award the 3 prizes to the 5 students.

Permutations – The question tests the understanding of permutations, specifically how to calculate the number of ways to arrange a subset of items (prizes) from a larger set (students).
Combinatorial Counting – It also involves combinatorial counting principles, where the order of selection matters since the prizes are different.